Open problems raised during the Workshop in Analysis and Probability

July-August 2009

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Problem lists from previous years: 2008 | 2007 | Older 1 | Older 2

The problems here were either submitted specifically for the purpose of inclusion in this list, or were taken from talks given during the Workshop in Linear Analysis and Probability.

Problem 1 (Submitted by Francisco Garcia) Let

be a normed space.

(a): Suppose is strictly convex. Can one renorm it so that its completion is strictly convex?
(b): Suppose is locally uniformly convex. Can one renorm it so that its completion is locally uniformly convex?

Problem 2 (Submitted by Mrinal Raghupathi) Let

be a finite Blaschke product, $z_1,\dotsc,z_n \in \mathbb{D}$ and

the ideal $\{ f\in H^\infty : f(z_1)=\cdots=f(z_n)=0 \}$ . What is the least dimension of a Hilbert space on which the quotient $({\mathbb{C}}+ B\cdot H^\infty)/I$ can be represented isometrically? In particular, are there finite dimensional ones?

Problem 3 (Submitted by Yun-Su Kim) Does the Riesz representation theorem hold for Hilbert spaces with respect to

-valued norms?

Problem 4 (Submitted by Yun-Su Kim) Is every Banach space a Hilbert space with respect to some

-valued norm?

Problem 5 (Submitted by Greg Kuperberg) The Mahler conjecture asserts that the product volume $({\mathrm{Vol}}K)({\mathrm{Vol}}K^\circ)$ is minimized for Hanner polytopes. Is the volume of the starlike body $K^\diamond$ defined in my paper maximized for Hanner polytopes?

Problem 6 (Submitted by Greg Kuperberg)

(a): Is the variance $V_{K \times K^\circ} [x\cdot y]$ maximized by ellipsoids?
(b): In particular, is this conjecture easier for log-concave or -concave measures in one dimension?